PPT-Section 9.4 Infinite Series: “Convergence Tests”

All graphics are attributed to Calculus10E by Howard Anton Irl Bivens and Stephen Davis Copyright 2009 by John Wiley amp Sons Inc All rights reserved Introduction In the last section we showed how to find the sum of a series by finding a closed form for the nth partial sum and taking its limit. DEFINITION. Absolute . Convergence. Verify that the series. converges absolutely.. This series converges absolutely because the positive series . (. with . absolute values. ) is a . p. -series with . Section 8.3b. Sometimes we cannot evaluate an improper. i. ntegral directly .  In these cases, we first try to. d. etermine whether it converges or diverges.. Diverges???... End of story, we’re done.. Convergence . Tests and Taylor . Series. Part I: Convergence Tests. Objectives. Know how to decide if a series converges or not. Corresponding sections in Simmons: 13.5, 13.6,13.7,13.8. Important examples. Section 9.4b. Recall the Direct Comparison Test from last class…. To apply this test, the terms of the unknown series must be. nonnegative. . This doesn’t limit the usefulness of this test,. because we can apply it to the absolute value of the series.. Series. Find sums of infinite geometric series.. Use mathematical induction to prove statements.. Objectives. infinite geometric series. converge. limit. diverge. mathematical induction. Vocabulary. In Lesson 12-4, you found partial sums of geometric series. You can also find the sums of some infinite geometric series. An . Section 9.4a. Consider the sentence. For what values of . x . is this an identity?. On the left is a function with domain of all real numbers,. and on the right is a limit of Taylor polynomials…. As we have already explored in previous sections (check the. Section 8.8 AP Calculus. Def Power Series. If x is a variable, then an infinite series of the form. is called a . power series. . . Generally,. is called a . power series centered at c. , where c is a constant.. Find the interval of convergence and the function of . x. r. epresented by the given geometric series.. This series will only converge when :. Interval of convergence:. Do Now: #18 and 20 on p.466. Objectives: You should be able to. …. Formulas. The goal in this section is to find the sum of an infinite geometric series. However, this objective is very closely connected to the limit of an infinite sequence. . A. finite . sum of real numbers always produces a real number,. but an . infinite. sum of real numbers is not actually a real sum:. Definition: Infinite Series. An . infinite series . is an expression of the form. Occasionally, a series may have both positive and negative terms and not be an alternating series. For instance, the series. has both positive and negative terms, yet it is not an alternating series. One way to obtain some information about the convergence of this series is to investigate the convergence of the series. Section 10.1. Sequences. Section 10.2. Infinite Series. Section 10.3. The Integral Test. 10.4. Comparison Tests. Section 10.5. Absolute Convergence; The . Ratio and Root Tests. Section 10.6. Alternating . All graphics are attributed to:. Calculus,10/E. by Howard Anton, Irl Bivens, and Stephen Davis. Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.”. Introduction. The purpose of this section is to discuss sums that contain infinitely many terms. Consider the following sequence . , . , . , . ,…. Each term of this sequence is of the form .  . What happens to these terms as n gets very large? . In general, the . , for all positive r .  . Many sequences have limiting factors.